Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,8,7)(10,12)(13,14,17,18)(15,16,19,20);(2,3)(5,6)(7,8);(2,3)(6,7)(14,18)(16,20);(9,10)(11,12)(13,15,17,19)(14,16,18,20);(1,3,4,2)(5,7,8,6)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^9|a^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2397, C2×#4306434, #72279775, … |
| maximal normal subgroup of | D4×#2397, C2×#4306434, #72279775, … |
| maximal quotient of | C2×#4306434, #72298306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213630);